# The Newtonian Revolution in Science

Last Updated August 12, 2024.

*The Newtonian Revolution,*Cambridge University Press, 1980, pp. 1-37, 290-99.

*[In the following essay, Cohen offers an overview of the developments in the scientific community during Newton's time. Cohen then identifies the qualities of Newton's* Principia *that made the work so revolutionary.]*

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1.1 Some basic features of the Scientifc Revolution
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A study of the Newtonian revolution in science rests on the fundamental assumption that revolutions actually occur in science. A correlative assumption must be that the achievements of Isaac Newton were of such a kind or magnitude as to constitute a revolution that may be set apart from other scientific revolutions of the sixteenth and seventeenth centuries. At once we are apt to be plunged deep into controversy. Although few expressions are more commonly used in writing about science than "scientific revolution", there is a continuing debate as to the propriety of applying the concept and term "revolution" to scientific change.' There is, furthermore, a wide difference of opinion as to what may constitute a revolution. And although almost all historians would agree that a genuine alteration of an exceptionally radical nature (*the* Scientific Revolution^{2}) occurred in the sciences at some time between the late fifteenth (or early sixteenth) century and the end of the seventeenth century, the question of exactly when this revolution occurred arouses as much scholarly disagreement as the cognate question of precisely what it was. Some scholars would place its origins in 1543, the year of publication of both Vesalius's great work on the fabric of the human body and Copernicus's treatise on the revolutions of the celestial spheres (Copernicus, 1543; Vesalius, 1543). Others would have the revolution be inaugurated by Galileo, possibly in concert with Kepler, while yet others would see Descartes as the true prime revolutionary. Contrariwise, a whole school of historians declare that many of the most significant features of the so-called Galilean revolution had emerged during the late Middle Ages.^{3}

A historical analysis of the Newtonian revolution in science does not, however, require participation in the current philosophical and sociological debates on these issues. For the fact of the matter is that the concept of revolution in science—in the sense in which we would understand this term nowadays—arose during Newton's day and was applied (see §2.2) first to a part of mathematics in which he made his greatest contribution, the calculus, and then to his work in celestial mechanics. Accordingly, the historian's task may legitimately be restricted to determining what features of Newton's science seemed so extraordinary in the age of Newton as to earn the designation of revolution. There is no necessity to inquire here into the various meanings of the term "revolution" and to adjudge on the basis of each such meaning the correctness of referring to a Newtonian revolution in the sciences.

The new science that took form during the seventeenth century may be distinguished by both external and internal criteria from the science and the philosophical study or contemplation of nature of the antecedent periods. Such an external criterion is the emergence in the seventeenth century of a scientific "community": individuals linked together by more or less common aims and methods, and dedicated to the finding of new knowledge about the external world of nature and of man that would be consonant with—and, accordingly, testable by—experience in the form of direct experiment and controlled observation. The existence of such a scientific community was characterized by the organization of scientific men into permanent formal societies, chiefly along national lines, with some degree of patronage or support by the state.^{4} The primary goal of such societies was the improvement of "natural knowledge".^{5} One way by which they sought to gain that end was through communication; thus the seventeenth century witnessed the establishment of scientific and learned journals, often the organs of scientific societies, including the *Philosophical Transactions* of the Royal Society of London, the *Journal des Sçavans,* and the *Acta eruditorum* of Leipzig.^{6} Another visible sign of the existence of a "new science" was the founding of research institutions, such as the Royal Greenwich Observatory, which celebrated its three-hundredth birthday in 1975. Newton's scientific career exhibits aspects of these several manifestations of the new science and the scientific community. He depended on the Astronomer Royal, John Flamsteed, for observational evidence that Jupiter might perturb the orbital motion of Saturn near conjunction and later needed lunar positions from Flamsteed at the Greenwich Observatory in order to test and to advance his lunar theory, especially in the 1690s. His first scientific publication was his famous article on light and colors, which appeared in the pages of the *Philosophical Transactions;* his *Principia* was officially published by the Royal Society, of which he became president in 1703 (an office he kept until his death in 1727). While the Royal Society was thus of great importance in Newton's scientific life, it cannot be said that his activities in relation to that organization or its journal were in any way revolutionary.

The signs of the revolution can also be seen in internal aspects of science: aims, methods, results. Bacon and Descartes agreed on one aim of the new science, that the fruits of scientific investigation would be the improvement of man's condition here on earth:^{7} agriculture, medicine, navigation and transportation, communication, warfare, manufacturing, mining.' Many scientists of the seventeenth century held to an older point of view, that the pursuit of scientific understanding of nature was practical insofar as it might advance man's comprehension of the divine wisdom and power. Science was traditionally practical in serving the cause of religion; but a revolutionary feature of the new science was the additional pragmatic goal of bettering everyday life here and now through applied science. The conviction that had been developing in the sixteenth and seventeenth centuries, that a true goal of the search for scientific truth must be to affect the material conditions of life, was then strong and widely shared, and constituted a novel and even a characteristic feature of the new science.

Newton often declared his conviction as to the older of these practicalities, as when he wrote to Bentley about his satisfaction in having advanced the cause of true religion by his scientific discoveries. Five years after the publication of his *Principia,* he wrote to Bentley that while composing the *Principia* ('my Treatise about our system'), 'I had an eye upon such Principles as might work with considering Men, for the Belief of a Deity' (Newton, 1958, p. 280; 1959-1977, vol. 3, p. 233). About two decades later, in 1713, he declared in the concluding general scholium to the *Principia* that the system of the world 'could not have arisen without the design and dominion of an intelligent and powerful being'. Newton was probably committed to some degree to the new practicality; at least he served as advisor to the official group concerned with the problem of finding methods of determining the longitude at sea. Yet it was not Newton himself, but other scientists such as Halley, who attempted to link the Newtonian lunar theory with the needs of navigators, and the only major practical innovation that he produced was an instrument for science (the reflecting telescope) rather than inventions for man's more mundane needs.^{9}

Another feature of the revolution was the attention to method. The attempts to codify method—by such diverse figures as Descartes, Bacon, Huygens, Hooke, Boyle, and Newton—signify that discoveries were to be made by applying a new tool of inquiry (a *novum organum,* as Bacon put it) that would direct the mind unerringly to the uncovering of nature's secrets. The new method was largely experimental, and has been said to have been based on induction; it also was quantitative and not merely observational and so could lead to mathematical laws and principles. I believe that the seventeenth-century evaluation of the importance of method was directly related to the role of experience (experiment and observation) in the new science. For it seems to have been a tacit postulate that any reasonably skilled man or woman should be able to reproduce an experiment or observation, provided that the report of that experiment or observation was given honestly and in sufficient detail. A consequence of this postulate was that anyone who understood the true methods of scientific enquiry and had acquired the necessary skill to make experiments and observations could have made the discovery in the first instance-provided, of course, that he had been gifted with the wit and insight to ask the right questions.^{10}

This experimental or experiential feature of the new science shows itself also in the habit that arose of beginning an enquiry by repeating or reproducing an experiment or observation that had come to one's attention through a rumor or an oral or written report. When Galileo heard of a Dutch optical invention that enabled an observer to see distant objects as clearly as if they were close at hand, he at once set himself to reconstructing such an instrument." Newton relates how he had bought a prism 'to try therewith the celebrated Phaenomena of Colours'.^{12} From that day to this, woe betide any investigator whose experiments and observations could not be reproduced, or which were reported falsely; this attitude was based upon a fundamental conviction that nature's occurrences are constant and reproducible, thus subject to universal laws. This twin requirement of performability and reproducibility imposed a code of honesty and integrity upon the scientific community that is itself yet another distinguishing feature of the new science.

The empirical aspect of the new science was just as significant with respect to the result achieved as with respect to the aims and methods. The law of falling bodies, put forth by Galileo, describes how real bodies actually fall on this earth—due consideration being given to the difference between the ideal case of a vacuum and the realities of an air-filled world, with winds, air resistance, and the effects of spin. Some of the laws of uniform and accelerated motion announced by Galileo can be found in the writings of certain late medieval philosopher-scientists, but the latter (with a single known exception of no real importance^{13}) never even asked whether these laws might possibly correspond to any real or observable motions in the external world. In the new science, laws which do not apply to the world of observation and experiment could have no real significance, save as mathematical exercises. This point of view is clearly enunciated by Galileo in the introduction of the subject of 'naturally accelerated motion', in his *Two New Sciences* (1638). Galileo states the aim of his research to have been 'to seek out and clarify the definition that best agrees with that [accelerated motion] which nature employs' (Galileo, 1974, p. 153; 1890-1909, vol. 8, p. 197). From his point of view, there is nothing 'wrong with inventing at pleasure some kind of motion and theorizing about its consequent properties, in the way that some men have derived spiral and conchoidal lines from certain motions, though nature makes no use of these [paths]'. But this is different from studying motion in nature, for in exploring phenomena of the real external world, a definition is to be sought that accords with nature as revealed by experience:

But since nature does employ a certain kind of acceleration for descending heavy things, we decided to look into their properties so that we might be sure that the definition of accelerated motion which we are about to adduce agrees with the essence of naturally accelerated motion. And at length, after continual agitation of mind, we are confident that this has been found, chiefly for the very powerful reason that the essentials successively demonstrated by us correspond to, and are seen to be in agreement with, that which physical experiments

[naturalia experimenta]show forth to the senses [ibid.].

Galileo's procedure is likened by him to having 'been led by the hand to the investigation of naturally accelerated motion by consideration of the custom and procedure of nature herself.

Like Galileo, Newton the physicist saw the primary importance of concepts and rules or laws that relate to (or arise directly from) experience. But Newton the mathematician could not help but be interested in other possibilities. Recognizing that certain relations are of physical significance (as that 'the periodic times are as the 3/2 power of the radii', or Kepler's third law), his mind leaped at once to the more universal condition (as that 'the periodic time is as any power *R*^{n} of the radius R').^{14} Though Newton was willing to explore the mathematical consequences of attractions of spheres according to any rational function of the distance, he concentrated on the powers of index 1 and -2 since they are the ones that occur in nature: the power of index 1 of the distance from the center applies to a particle within a solid sphere and the power of index -2 to a particle outside either a hollow or solid sphere.^{15} It was his aim, in the *Principia*, to show that the abstract or 'mathematical principles' of the first two books could be applied to the phenomenologically revealed world, an assignment which he undertook in the third book. To do so, after Galileo, Kepler, Descartes, and Huygens, was not in itself revolutionary, although the scope of the *Principia* and the degree of confirmed application could well be so designated and thus be integral to the Newtonian revolution in science.

An excessive insistence on an out-and-out empirical foundation of seventeenth-century science has often led scholars to exaggerations.^{16} The scientists of that age did not demand that each and every statement be put to the test of experiment or observation, or even have such a capability, a condition that would effectively have blocked the production of scientific knowledge as we know it. But there was an insistence that the goal of science was to understand the real external world, and that this required the possibility of predicting testable results and retrodicting the data of actual experience: the accumulated results of experiment and controlled observation. This continual growth of factual knowledge garnered from the researches and observations made all over the world, paralleled by an equal and continual advance of understanding, was another major aspect of the new science, and has been a distinguishing characteristic of the whole scientific enterprise ever since. Newton certainly made great additions to the stock of knowledge. In the variety and fundamental quality of these contributions we may see the distinguishing mark of his great creative genius, but this is something distinct from having created a revolution.

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1.2 A Newtonian revolution in science: the varieties of Newtonian science
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In the sciences, Newton is known for his contributions to pure and applied mathematics, his work in the general area of optics, his experiments and speculations relating to theory of matter and chemistry (including alchemy), and his systematization of rational mechanics (dynamics) and his celestial dynamics (including the Newtonian "system of the world"). Even a modest portion of these achievements would have sufficed to earn him an unquestioned place among the scientific immortals. In his own day (as we shall see below in Ch. 2), the word "revolution" began to be applied to the sciences in the sense of a radical change; one of the first areas in which such a revolution was seen to have occurred was in the discovery or invention of the calculus: a revolution in mathematics.' There is also evidence aplenty that in the age of Newton and afterwards, his *Principia* was conceived to have ushered in a revolution in the physical sciences. And it is precisely this revolution whose characteristic features I aim to elucidate.

Newton's studies of chemistry and theory of matter yielded certain useful results^{2} and numerous speculations. The latter were chiefly revealed in the queries at the end of the *Opticks,* especially the later ones,^{3} and in such a tract as the *De natura acidorum.*^{4} The significance of these writings and their influence have been aggrandized (from Newton's day to ours) by the extraordinary place in science held by their author. At best, they are incomplete and programmatic and—in a sense—they chart out a possible revolution, but a revolution never achieved by Newton nor ever realized along the lines that he set down. Newton's program and suggestions had a notable influence on the science of the eighteenth century, particularly the development of theories of heat and electricity (with their subtle elastic fluids) (cf. Cohen, 1956, Ch. 7, 8). Newton had a number of brilliant insights into the structure of matter and the process of chemical reaction, but the true revolution in chemistry did not come into being until the work of Lavoisier, which was not directly Newtonian (see Guerlac, 1975).

The main thrust of Newton's views on matter was the hope of deriving 'the rest of the phenomena of nature by the same kind of reasoning from mechanical principles' that had served in deducing 'the motions of the planets, the comets, the moon, and the sea'. He was convinced that all such phenomena, as he said in the preface (1686) to the first edition of the *Principia,* 'may depend upon certain forces by which the particles of bodies … are either mutually impelled [attracted] toward one another so as to cohere in regular figures' or 'are repelled and recede from one another'.^{5} In this way, as he put it on another occasion, the analogy of nature would be complete: 'Whatever reasoning holds for greater motions, should hold for lesser ones as well. The former depend upon the greater attractive forces of larger bodies, and I suspect that the latter depend upon the lesser forces, as yet unobserved, of insensible particles'. In short, Newton would have nature be thus 'exceedingly simple and conformable to herself.^{6} This particular program was a conspicuous failure. Yet it was novel and can even be said to have had revolutionary features, so that it may at best represent a failed (or at least a never-achieved) revolution. But since we are concerned here with a positive Newtonian revolution, Newton's hope to develop a micro-mechanics analogous to his successful macromechanics is not our main concern. We cannot wholly neglect this topic, however, since it has been alleged that Newton's mode of attack on the physics of gross bodies and his supreme success in celestial mechanics was the product of his investigations of short-range forces, despite the fact that Newton himself said (and said repeatedly) that it was his success in the area of gravitation that led him to believe that the forces of particles could be developed in the same style. R. S. Westfall (1972, 1975) would not even stop there, but would have the 'forces of attraction between particles of matter', and also 'gravitational attraction which was probably the last one [of such forces] to appear', be 'primarily the offspring of alchemical active principles'. This particular thesis is intriguing in that it would give a unity to Newton's intellectual endeavor; but I do not believe it can be established by direct evidence (see Whiteside, 1977). In any event, Newton's unpublished papers on alchemy and his published (and unpublished) papers on chemistry and theory of matter hardly merit the appellation of "revolution", in the sense of the radical influence on the advance of science that was exerted by the *Principia.*

In optics, the science of light and colors, Newton's contributions were outstanding. But his published work on 'The Reflections, Refractions, Inflexions [i.e., diffraction] & Colours of Light', as the *Opticks* was subtitled, was not revolutionary in the sense that the *Principia* was. Perhaps this was a result of the fact that the papers and book on optics published by Newton in his lifetime do not boldly display the mathematical properties of forces acting (as he thought) in the production of dispersion and other optical phenomena, although a hint of a mathematical model in the Newtonian style is given in passing in the *Opticks* (see §3.11) and a model is developed more fully in sect. 14 of bk. one of the *Principia.* Newton's first published paper was on optics, specifically on his prismatic experiments relating to dispersion and the composition of sunlight and the nature of color. These results were expanded in his *Opticks* (1704; Latin ed. 1706; second English ed. 1717/1718), which also contains his experiments and conclusions on other aspects of optics, including a large variety of what are known today as diffraction and interference phenomena (some of which Newton called the "inflexion" of light). By quantitative experiment and measurement he explored the cause of the rainbow, the formation of "Newton's rings" in sunlight and in monochromatic light, the colors and other phenomena produced by thin and thick "plates", and a host of other optical effects.^{7} He explained how bodies exhibit colors in relation to the type of illumination and their selective powers of absorption and transmission or reflection of different colors. The *Opticks,* even apart from the queries, is a brilliant display of the experimenter's art, where (as Andrade, 1947, p. 12, put it so well) we may see Newton's 'pleasure in shaping'. Some of his measurements were so precise that a century later they yielded to Thomas Young the correct values, to within less than I percent, of the wave lengths of light of different colors.^{8} Often cited as a model of how to perform quantitative experiments and how to analyze a difficult problem by experiment,^{9} Newton's studies of light and color and his *Opticks* nevertheless did not create a revolution and were not ever considered as revolutionary in the age of Newton or afterwards. In this sense, the *Opticks* was not epochal.

From the point of view of the Newtonian revolution in science, however, there is one very significant aspect of the *Opticks:* the fact that in it Newton developed the most complete public statement he ever made of his philosophy of science or of his conception of the experimental scientific method. This methodological declaration has, in fact, been a source of some confusion ever since, because it has been read as if it applies to all of Newton's work, including the *Principia.*^{10} The final paragraph of qu. 28 of the *Opticks* begins by discussing the rejection of any 'dense Fluid' supposed to fill all space, and then castigates 'Later Philosophers' (i.e., Cartesians and Leibnizians) for 'feigning Hypotheses for explaining all things mechanically, and referring other Causes to Metaphysicks'. Newton asserts, however, that 'the main Business of natural Philosophy is to argue from Phaenomena without feigning Hypotheses, and to deduce Causes from Effects, till we come to the very first Cause, which certainly is not mechanical'." Not only is the main assignment 'to unfold the Mechanism of the World', but it is to 'resolve' such questions as: 'What is there in places almost empty of Matter … ?' 'Whence is it that Nature doth nothing in vain; and whence arises all that Order and Beauty which we see in the World?' What 'hinders the fix'd Stars from falling upon one another?' 'Was the Eye contrived without Skill in Opticks, and the Ear without Knowledge of Sounds?' or 'How do the Motions of the Body follow from the Will, and whence is the Instinct in Animals?'

In qu. 31, Newton states his general principles of analysis and synthesis, or resolution and composition, and the method of induction:

As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy. And although the arguing from Experiments and Observations by Induction be no Demonstration of general Conclusions; yet it is the best way of arguing which the Nature of Things admits of, and may be looked upon as so much the stronger, by how much the Induction is more general.

Analysis thus enables us to

proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general.

This method of analysis is then compared to synthesis or composition:

And the Synthesis consists in assuming the Causes discover'd, and establish'd as Principles, and by them explaining the Phaenomena proceeding from them, and proving the Explanations.

^{12}

The lengthy paragraph embodying the foregoing three extracts is one of the most often quoted statements made by Newton, rivaled only by the concluding General Scholium of the *Principia,* with its noted expression: *Hypotheses non fingo.*

Newton would have us believe that he had himself followed this "scenario":^{13} first, to reveal by "analysis" some simple results that were generalized by induction, thus proceeding from effects to causes and from particular causes to general causes; then, on the basis of these causes considered as principles, to explain by "synthesis" the phenomena of observation and experiment that may be derived or deduced from them, 'proving the Explanations'. Of the latter, Newton says that he has given an 'Instance … in the End of the first Book' where the 'Discoveries being proved [by experiment] may be assumed in the Method of Composition for explaining the Phaenomena arising from them'. An example, occurring at the end of bk. one, pt. 2, is props. 8-11, with which pt.2 concludes. Prop. 8 reads: 'By the discovered Properties of Light to explain the Colours made by Prisms'. Props. 9-10 also begin: 'By the discovered Properties of Light to explain …', followed (prop. 9) by 'the Rain-bow' and (prop. 10) by 'the permanent Colours of Natural Bodies'. Then, the concluding prop. 11 reads: 'By mixing coloured Lights to compound a beam of Light of the same Colour and Nature with a beam of the Sun's direct Light'.

The formal appearance of the *Opticks* might have suggested that it was a book of synthesis, rather than analysis, since it begins (bk. one, pt. 1) with a set of eight 'definitions' followed by eight 'axioms'. But the elucidation of the propositions that follow does not make explicit reference to these axioms, and many of the individual propositions are established by a method plainly labeled 'The PROOF by Experiments'. Newton himself states clearly at the end of the final qu. 31 that in bks. one and two he has 'proceeded by … Analysis' and that in bk. three (apart from the queries) he has 'only begun the Analysis'. The structure of the *Opticks* is superficially similar to that of the *Principia,* for the *Principia* also starts out with a set of 'definitions' (again eight in number), followed by three 'axioms' (three 'axiomata sive leges motus'), upon which the propositions of the first two books are to be constructed (as in the model of Euclid's geometry). But then, in bk. three of the *Principia,* on the system of the world, an ancillary set of so-called 'phenomena' mediate the application of the mathematical results of bks. one and two to the motions and properties of the physical universe.^{14} Unlike the *Opticks,* the *Principia* does make use of the axioms and definitions.^{15} The confusing aspect of Newton's stated method of analysis and synthesis (or composition) in qu. 31 of the *Opticks* is that it is introduced by the sentence 'As in Mathematicks, so in Natural Philosophy …', which was present when this query first appeared (as qu. 23) in the Latin *Optice* in 1706, 'Quemadmodum in Mathematica, ita etiam in Physica …'. A careful study, however, shows that Newton's usage in experimental natural philosophy is just the reverse of the way "analysis" and "synthesis" (or "resolution" and "composition") have been traditionally used in relation to mathematics, and hence in the *Principia*—an aspect of Newton's philosophy of science that was fully understood by Dugald Stewart a century and a half ago but that has not been grasped by present-day commentators on Newton's scientific method, who would even see in the *Opticks* the same style that is to be found in the *Principia'*^{6} (this point is discussed further in §3.11).

Newton's "method", as extracted from his *dicta* rather than his *opera,* has been summarized as follows: 'The main features of Newton's method, it seems, are: The rejection of hypotheses, the stress upon induction, the working sequence (induction precedes deduction), and the inclusion of metaphysical arguments in physics' (Turbayne, 1962, p. 45). Thus Colin Turbayne would have 'the deductive procedure' be a defining feature of Newton's 'mathematical way' and Descartes's *'more geometrico'* respectively: 'Descartes's "long chains of reasoning" were deductively linked. Newton's demonstrations were reduced to "the form of propositions in the mathematical way"'. He would criticize those analysts who have not recognized that the defining property of 'the Cartesian "geometrical method" or the Newtonian "mathematical way"'—paradoxical as it may seem—need be neither geometrical nor mathematical. Its defining property is demonstration, not the nature of the terms used in it'.^{17} It may be observed that the phrase used here, 'the Newtonian "mathematical way"', or 'Newton's "mathematical way"', so often quoted in philosophical or methodological accounts of Newton's science, comes from the English translation of Newton's *System of the World*^{18} but is not to be found in any of the manuscript versions of that tract, including the one that is still preserved among Newton's papers (see Dundon, 1969; Cohen, 1969*a*, 1969*c*).

The Newtonian revolution in the sciences, however, did not consist of his use of deductive reasoning, nor of a merely external form of argument that was presented as a series of demonstrations from first principles or axioms. Newton's outstanding achievement was to show how to introduce mathematical analysis into the study of nature in a rather new and particularly fruitful way, so as to disclose *Mathematical Principles of Natural Philosophy,* as the *Principia* was titled in full: *Philosophiae naturalis principia mathematica.* Not only did Newton exhibit a powerful means of applying mathematics to nature, he made use of a new mathematics which he himself had been forging and which may be hidden from a superficial observer by the external mask of what appears to be an example of geometry in the traditional Greek style (see n. 10 to §1.3).

In the *Principia* the science of motion is developed in a way that I have characterized as the Newtonian style. In Ch. 3 it shall be seen that this style consists of an interplay between the simplification and idealization of situations occurring in nature and their analogues in the mathematical domain. In this manner Newton was able to produce a mathematical system and mathematical principles that could then be applied to natural philosophy, that is, to the system of the world and its rules and data as determined by experience. This style permitted Newton to treat problems in the exact sciences as if they were exercises in pure mathematics and to link experiment and observation to mathematics in a notably fruitful manner. The Newtonian style also made it possible to put to one side, and to treat as an independent question, the problem of the cause of universal gravity and the manner of its action and transmission.

The Newtonian revolution in the sciences was wrought by and revealed in the *Principia.* For more than two centuries, this book set the standard against which all other science was measured; it became the goal toward which scientists in such diverse fields as paleontology, statistics, and biochemistry would strive in order to bring their own fields to a desired high estate.^{19} Accordingly, I have striven in the following pages to explore and to make precise the qualities of Newton's *Principia* that made it so revolutionary. Chief among them, as I see it, is the Newtonian style, a clearly thought out procedure for combining mathematical methods with the results of experiment and observation in a way that has been more or less followed by exact scientists ever since. This study concentrates mainly on the *Principia,* because of the supreme and unique importance of that treatise in the Scientific Revolution and in the intellectual history of man. In the *Principia* the role of induction is minimal and there is hardly a trace of that analysis which Newton said should always precede synthesis.^{20} Nor is there any real evidence whatever that Newton first discovered the major propositions of the *Principia* in any way significantly different from the way in which they are published with their demonstrations.^{21} Newton's studies of optical phenomena, chemistry, theory of matter, physiological and sensational psychology, and other areas of experimental philosophy did not successfully exhibit the Newtonian style. Of course, whatever Newton said about method, or induction, or analysis and synthesis, or the proper role of hypotheses, took on an added significance because of the commanding scientific position of the author. This position was attained as a result of the revolution in science that, in the age of Newton (and afterwards), was conceived to be centered in his mathematical principles of natural philosophy and his system of the world (see Ch. 2). The general philosophical issues of induction, and of analysis and synthesis, gained their importance after Newton had displayed the system of the world governed by universal gravity, but they played no significant role in the way the Newtonian style is used in the elaboration of that system or in the disclosure of that universal force.

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1.3 Mathematics in the new science (1): a world of numbers
**

After modern science had emerged from the crucible of the Scientific Revolution, a characteristic expression of one aspect of it was given by Stephen Hales, often called the founder of plant physiology.' An Anglican clergyman and an ardent Newtonian, Hales wrote (1727) that 'we are assured that the all-wise Creator has observed the most exact proportions, of *number, weight and measure,* in the make of all things'; accordingly, 'the most likely way … to get any insight into the nature of those parts of the creation, which come within our observation, must in all reason be to number, weigh and measure' (Hales, 1969, p. xxxi). The two major subjects to which Hales applied this rule were plant and animal physiology: specifically the measurement of root and sap pressures in different plants under a variety of conditions and the measurement of blood pressure in animals. Hales called his method of enquiry 'statical', from the Latin version of the Greek word for weighing—in the sense that appears to have been introduced into the scientific thought of the West by Nicolaus Cusanus in the fifteenth century, in a treatise entitled *De staticis experimentis* (cf. Guerlac, 1972, p. 37; and Viets, 1922).

In the seventeenth century two famous examples of this 'statical' method were Santorio's experiments on the changes in weight that occurred in the daily life cycle of man (Grmek, 1975), and Van Helmont's experiment on the willow tree. The latter consisted of filling an earthen pot with a weighed quantity of soil that had been dried in a furnace, in which Helmont planted a previously weighed 'Trunk or Stem' of a willow tree. He 'covered the lip or mouth of the Vessel with an Iron plate covered with Tin', so that the dust flying about should not be 'co-mingled with the Earth' inside the vessel. He watered the earth regularly with rain water or distilled water for five years, and found that the original tree, weighing 5 pounds, now had grown to a weight of' 169 pounds, and about three ounces' (ignoring 'the weight of the leaves that fell off in the four Automnes'). Since the earth in the vessel, when dried out at the end of the experiment, was only 'about two ounces' less in weight than the original weight of 200 pounds, Helmont concluded that 164 pounds of 'Wood, Barks, and Roots' must have been formed out of water alone.^{2} Helmont did not know (or suspect) that the air itself might supply some of the weight of the tree, a discovery made by Hales, who repeated Helmont's experiment with the added precision of weighing the water added to the plant and measuring the plant's rate of 'perspiration' (Hales, 1969, Ch. 1, expts. 1-5). The original of this experiment had been proposed by Cusanus, but there is no certainty as to whether or not he may have actually performed it.

I have purposely chosen these first examples from the life (or biological) sciences, since it is usually supposed that in the Scientific Revolution, numerical reasoning was the prerogative of the physical sciences. One of the most famous uses of numerical reasoning in the Scientific Revolution occurs in Harvey's analysis of the movement of the blood. A central argument in Harvey's demonstration of the circulation is quantitative, based on an estimate of the capacity of the human heart; the left ventricle, he finds, when full may contain 'either 2, or 3, or *VA* oz.; I have found in a dead man above 4 oz.' Knowing that 'the heart in one half hour makes above a thousand pulses, yea in some, and at some times, two, three or four thousand', simple calculation shows how much blood the heart discharges into the arteries in a half hour—at least 83 pounds 4 ounces, 'which is a greater quantity than is found in the whole body'. He made similar calculations for a dog and a sheep. These numbers showed 'that more blood is continually transmitted through the heart, than either the food which we receive can furnish, or is possible in the veins'.^{3} Here we may see how numerical calculation provided an argument in support of theory: an excellent example of how numbers entered theoretical discussions in the new science.

Despite the force of the foregoing examples, however, it remains true that the major use of numerical reasoning in the science of the seventeenth century occurred in the exact physical sciences: optics, statics, kinematics and dynamics, astronomy, and parts of chemistry.^{4} Numerical relations of a special kind tended to become all the more prominent in seventeenth-century exact science because at that time the laws of science were not yet written in equations. Thus we today would write Galileo's laws of uniformly accelerated motion as *v* = *At*, and *s* = ½ *At*^{2}, but he expressed the essence tended of naturally accelerated motion (free fall, for example, or motion along an inclined plane) in language that sounds much more like number theory than like algebra: 'the spaces run through in equal times by a moveable descending from rest maintain among themselves the same rule *[rationem]* as do the odd numbers following upon unity'.^{5} Galileo's rule, that these first differences (or 'the progression of spaces') accord with the odd numbers, led him to another form of his rule, that the 'spaces run through in any times whatever' by a uniformly accelerated body starting from rest 'are to each other in the doubled ratio of the times [or; as the square of the times]' in which such spaces are traversed. This form of his rule, expressed in the language of ratios, comes closer to our own algebraic expression.^{6} Thus while speeds increase with time according to the natural numbers, total distances or spaces traversed increase (depending on the chosen measure) according to the odd numbers or the squares^{7} of the natural numbers.^{8}

In the exact science of the seventeenth century, considerations of shape, or of geometry, are to be found alongside rules of numbers. In a famous statement about the mathematics of nature, Galileo said:

Philosophy [i.e., natural philosophy, or science] is written in that vast book which stands forever open before our eyes, I mean the universe; but it cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

^{9}

This is not the philosophy of Newton, where mathematics suggests at once a set of equations or proportions (which may be verbal), infinite series, and the taking of limits.^{10} In fact, the above-quoted statement almost sounds like Kepler, rather than Galileo. It was Kepler who found in numerical geometry a reason why the Copernican system is to be preferred to the Ptolemaic. In one of these systems—the Ptolemaic—there are seven 'planets' or wanderers (sun and moon; Mercury and Venus; and Mars, Jupiter, and Saturn), but in the other there are only six planets (Mercury and Venus; the earth; and Mars, Jupiter, and Saturn). Suppose that each planet is associated with a giant spherical shell in which it moves (or which contains its orbit). Then there would be five spaces between each pair of successive spheres. Kepler knew of Euclid's proof that there are only five regular solids that can be constructed by simple geometrical rules (cube, tetrahedron, dodecahedron, icosahedron, octahedron). By choosing them in the above order, Kepler found that they would just fit into the spaces between the spheres of the planetary orbits, the only error of any consequence occurring in the case of Jupiter. Hence number and geometry showed that there must be six planets, as in the Copernican system, and not seven, as in the Ptolemaic.^{11}

Rheticus, Copernicus's first and only disciple, had proposed a purely numerical argument for the Copernican system. In the suncentered universe there are six planets, he said, and 6 is the first 'perfect' number (that is, it is the sum of its divisors, 6 = 1 + 2 + 3).^{12} Kepler, however, rejected the perfect-number argument of Rheticus, preferring to base his advocacy of the Copernican system on the five perfect solids. He said:

I undertake to prove that God, in creating the universe and regulating the order of the cosmos, had in view the five regular bodies of geometry as known since the days of Pythagoras and Plato, and that He has fixed, according to those dimensions, the number of heavens, their proportions, and the relations of their movements.

^{13}

It is not without interest, accordingly, that when Kepler heard that Galileo had discovered some new "planets" by using a telescope, he was greatly concerned lest his own argument should fall to the ground (cf. Kepler, 1965, p. 10). How happy he was, he recorded, when the "planets" discovered by Galileo turned out to be secondary and not primary "planets", that is, satellites of planets.

Two reactions to Galileo's discovery of four new "planets" may show us that the use of numbers in the exact sciences in the seventeenth century was very different from what we might otherwise have imagined. Francesco Sizi, in opposition to Galileo, argued that there must be seven and only seven "planets"; hence Galileo's discovery was illusory. His assertion about the number seven was based on its occurrence in a number of physical and physiological situations, among them the number of openings in the head (two ears, two eyes, two nostrils, one mouth).^{14} Kepler, who applauded Galileo, proposed that he look next for the satellites of Mars and of Saturn, since the numerical sequence of the satellites (one for the earth and four for Jupiter) seemed to demand two for Mars and eight (or possibly six) for Saturn: 1, 2, 4, 8.^{15} This type of numerical reasoning had deleterious effects on astronomy in the case of at least one major scientist: Christiaan Huygens. For when Huygens had discovered a satellite of Saturn, he did not bother to look for any further ones. He was convinced, as he boldly declared in the preface to his *Systema Saturnium* (1659), that there could be no others (Huygens, 1888-1950, vol. 15, pp. 212sq). With his discovery of a new satellite, he said, the system of the universe was complete and symmetrical: one and the same "perfect" number 6 in the primary planets and in the secondary "planets" (or planetary satellites). Since his telescope could resolve the ring of Saturn and solve the mystery of its strange and inconstant shape, it could have revealed more satellites had Huygens not concluded that God had created the universe in two sets of planetary bodies, six to a set, according to the principle of "perfect" numbers.'" Such examples all illustrate some varieties of the association of numbers with actual observations. That we today would not accept such arguments is probably less significant than the fact that those who did included some major founders of our modern science, among them Kepler and Huygens, and Cassini.^{17}

**
1.4 Mathematics in the new science (2): exact laws of nature and the hierarchy of causes
**

In addition to the search for special numbers (odd, prime, perfect, the number of regular solids), which did not always lead to useful results, the scientists of the seventeenth century—like scientists ever since—also sought exact relations between the numbers obtained from measurement, experiment, and observation. An example is Kepler's third (or "harmonic") law. In the Copernican system, each of the planets has a speed that seems related to its distance from the sun: the farther from the sun, the slower the speed. Both Galileo and Kepler were convinced that the speeds and distances could not be arbitrary; there must be some exact relation between these two quantities; for God, in creating the universe, must have had a plan, a law. The Keplerian scheme of the five regular solids imbedded in a nest of spheres showed an aspect of mathematical "necessity" in the distribution of the planets in space, but it did not include the data on their speeds. Thus it only partly satisfied Kepler's goal as a Copernican, expressed by him as follows: 'There were three things in particular, namely, the number, distances and motions of the heavenly bodies, as to which I [Kepler] searched zealously for reasons why they were as they were and not otherwise."^{1}

In the *Mysterium cosmographicum* (1596), in which he had used the five regular solids to show why there were five and only five planets spaced as in the Copernican system, Kepler had also tried to find 'the proportions of the motions [of the planets] to the orbits'. The orbital speed of a planet depends upon its average distance from the sun (and hence the circumference of the orbit) and its sidereal period of revolution, both values given by Copernicus in his *De revolutionibus* (1543) with a reasonably high degree of accuracy. Kepler decided that the 'anima motrix' which acts on the planets loses its strength as the distance from the sun gets greater. But rather than assuming that this force diminishes as the square of the distance (which would mean that it spreads out uniformly in all directions, as light does), Kepler thought it more likely that this force would diminish in proportion to the circle or orbit over which it spreads, directly as the increase of the distance rather than as the square of the increase of the distance. The distance from the sun, according to Kepler, 'acts twice to increase the period' of a planet; for it acts once in slowing down the planet's motion, according to the law by which the force that moves the planet weakens in proportion as the distance increases, and again because the total path along which the planet has to move to complete a revolution increases as the distance from the sun increases. Or, 'conversely half the increase of the period is proportional to the increase of distance'.^{2} This relationship comes near to the truth, Kepler observes, but he sought in vain for more than two decades for an accurate relation between the average distance *(a)* of the planets and their periods *(T).* Eventually it occurred to him to use higher powers of *a* and *T,* and on 15 May 1618 he found that the 'periodic times of any two planets are in the sesquialteral [3/2] ratio of their mean distances', that is, the ratio of the squares of the periods is the same as the ratio of the cubes of their average distances, a relationship which we express as *a*^{3}/*T*^{2} = const. and call Kepler's third law.^{3} It should be noted that Kepler's discovery apparently resulted from a purely numerical exercise and insofar differed from his discovery of the area law and of the law of elliptic orbits, both of which were presented originally (and may have been discovered) in association with a definite causal concept of solar force and a principle of force and motion.^{4}

Galileo's approach to this problem was based on a kinematical law rather than on purely numerical considerations: the principle of naturally accelerated motion, which he had discovered in his studies of freely falling bodies.^{5} He thought so well of his solution to the cosmic problem that he introduced it into both his *Dialogo* (1632) or the *Dialogue Concerning the Two Chief World Systems,* and his *Discorsi* (1638), or the *Two New Sciences* (Galileo, 1953, pp. 29sq; 1890-1909, vol. 4, pp. 53sq; also 1974, pp. 232-234; 1890-1909, vol. 8, pp. 283sq). He attributed the basic idea to Plato, but there is nothing even remotely resembling it in any of the Platonic works, nor has this idea been found in any known Neoplatonic composition or commentary, ancient, medieval, or modern.^{6} Galileo said that there was a point out in space from which God had let fall all of the planets. When each planet arrived at its proper orbit, it would have attained its proper orbital speed and would have needed only to be tumed in its path to accord with the known values of planetary distances and speeds. Galileo did not specify where that point is located, and (as an analysis by Newton revealed) the point would in fact have to be infinitely far away.^{7} Galileo, furthermore, did not understand that such a descent toward the sun would require a constantly changing acceleration, which in dynamics would correspond to a constantly changing solar-planetary force that varies inversely as the square of the distance. In this example we may see that Galileo could have had no conception of a solar gravitating force. His discussion does not contain the slightest hint of a relation between force and acceleration that might be said to have contained a germ of Newton's second law of motion.^{8}

Galileo was primarily successful in applying mathematics to such areas as statics and kinematics, in both of which there is no need to take account of physical causes, such as quantifiable forces. As he himself says, in his *Two New Sciences:*

The present does not seem to me to be an opportune time to enter into the investigation of the cause of the acceleration of natural motion, concerning which various philosophers have produced various opinions.… For the present, it suffices … that we … investigate and demonstrate some attributes of a motion so accelerated (whatever be the cause of its acceleration) that the momenta of its speed go increasing … in that simple ratio with which the continuation of time increases … [Galileo, 1974, pp. 158sq; 1890-1909, vol. 8, p. 202].

In part, but only in part, his procedure resembles that of the late medieval kinematicists. Like them, he defines uniform motion and then proceeds to uniformly accelerated motion. Almost at once, he reveals the mean-speed law: In uniformly accelerated motion during a time *t* the distance traveled is the same as if there had been uniform motion at the mean value of the changing speeds during that same time (Galileo, 1974, p. 165; 1890-1909, vol. 8, p. 208). Since the motion is uniform, the mean value is one half of the sum of the initial and final speeds. If we may somewhat anachronistically translate Galileo's verbal statements of ratios into their equivalent equations, we may show that he has proved that *s* = *V̆t* where *V̆* = (*v*_{1}, + *v*_{2})/2. Since *v*_{2} = *v*_{1} + *At*, it would follow at once that *s* = *v*_{1}*t* + ½*At*^{2}. In the special case of motion starting from rest, *v*_{1} = 0 and *s* = ½*At*^{2}.

Thus far, except for the final result (in which the relation *s* = [(*v*_{1}, + *v*_{2})/2]*t* leads to *s* = *vt* + ½*At*^{2}), Galileo could be proceeding much like his fourteenth-century predecessors.^{9} But there are significant differences of such consequence that we may easily discern in Galileo's *Two New Sciences* the beginnings of our own science of motion, whereas this feature is lacking in the medieval treatises. The major difference is that the writers of the fourteenth century were not concerned with the physics of motion, with nature as revealed by experiment and observation. Thus they constructed a generalized "latitude of forms", a mathematico-logical analysis of any quality that can be quantified, of which motion (in the sense of "local" motion, from one place to another) is but one example, along with such other quantifiable qualities as love, virtue, grace, whiteness, hotness, and so on. Even in the case of motion, they were dealing with Aristotelian "motion", defined in very general terms as the transition from actuality to potentiality. For two centuries, there is no record of any scholar ever applying the principles of uniform and accelerated motion to actual motions as observed on earth or in the heavens. Prior to Galileo, only Domingo de Soto made such an application, and he appears as a *lusus naturae* of no real importance (see §1.1, n. 13).

How different it is with Galileo! He based his very definitions on nature herself. His aim was not to study motion in the abstract, but the observed motions of bodies. The true test of his mathematical laws (as *s* = ½*At*^{2}) was not their logical consistency but their conformity with the results of actual experimental tests. Thus much is said in the public record.^{10} But now we know additionally, thanks to the studies of Galileo's manuscripts by Stillman Drake, that Galileo was making experiments not only to relate his discovered laws to the world of nature, but also as part of the discovery process itself.

Thus in the exact sciences of the seventeenth century we may observe a hierarchy of mathematical laws. First, there are mathematical laws deduced from certain assumptions and definitions, and which lead to experimentally testable results. If, as in Galileo's case, the assumptions and definitions are consonant with nature, then the results should be verifiable by experience. When Galileo sets forth, as a postulate, that the speed acquired in naturally accelerated motion is the same along all planes of the same heights, whatever their inclination, he declares that the 'absolute truth' of this postulate 'will be later established for us by our seeing that other conclusions, built on this hypothesis, do indeed correspond with and exactly conform to experiment'. This reads like a classic statement of the hypothetico-deductive method; but it is to be observed that it is devoid of any reference to the physical nature of the cause of the acceleration. Such a level of discourse is not essentially different in its results from another seventeenth-century way of finding mathematical laws of nature without going into causes: by the direct analysis of the data of experiment and observation. We have seen this to have been in all probability Kepler's procedure in finding his third (or "harmonic") law of planetary motion. Other examples are Boyle's law of gases and Snel's law of refraction (see Mach, 1926, pp. 32-36; Sabra, 1967; Hoppe, 1926, pp. 33sq).

The second level of the hierarchy is to go beyond the mathematical description to some sort of causes. Boyle's law, for example, is a mathematical statement of proportionality between two variables, each of which is a physical entity related to an observable or measurable quantity. Thus the volume *(V)* of the confined gas is measured by the mercury level according to some volumetric scale, and the pressure of the confined gas is determined by the difference between two mercury levels *(h)* plus the height of the mercury column in a barometer (h,). Boyle's experiments showed that the product of L^{1} and *h + hi* is a constant. The sum *h* + *h* is a height (in inches) of a mercury column equivalent to a total pressure exerted on and by the confined gas; what is measured directly in this case is not the pressure but a quantity (height of mercury) which itself is a measure of (and so can stand for) pressure. But nothing is said concerning the cause of pressure in a confined gas, nor of the reason why this pressure should increase as the gas is confined into a smaller volume, a phenomenon known to Boyle before he undertook the experiments and which he called the "spring" of the air. Now the second level of hierarchy is to explore the cause of this "spring". Boyle suggested two physical models that might serve to explain this phenomenon. One is to think of each particle being itself compressible, in the manner of a coiled spring or a piece of wood, so that the air would be 'a heap of little bodies, lying one upon another, as may be resembled to a fleece of wool'. Another is to conceive that the particles are in constant agitation, in which case 'their elastical power is not made to depend upon their shape or structure, but upon the vehement agitation'. Boyle himself, on this occasion, did not choose to decide between these explanations or to propose any other (see Cohen, 1956, p. 103; Boyle, 1772, vol. 1, pp. llsq). But the example does show that in the exact or quantitative sciences of the seventeenth century, there was a carefully observed distinction between a purely mathematical statement of a law and a causal mechanism for explaining such a law, that is, between such a law as a mathematical description of phenomena and the mathematical and physical exploration of its cause.

In some cases, the exploration of the cause did not require such a mechanical model, or explanation of cause, as the two mentioned by Boyle. For example, the parabolic path of projectiles is a mathematical statement of a phenomenon, with the qualifications arising from air resistance. But the mathematical conditions of a parabola are themselves suggestive of causes: for—again with the qualifications arising from air resistance—they state that there is uniform motion in the horizontal component and accelerated motion in the downward component. Since gravity acts downward and has no influence in the horizontal component, the very mathematics of the situation may lead an inquirer toward the physical causes of uniform and accelerated motion in the parabolic path of projectiles. Similarly, Newton's exploration of the physical nature and cause of universal gravity was guided by the mathematical properties of this force: that it varies inversely as the square of the distance, is proportional to the masses of the gravitating bodies and not their surfaces, extends to vast distances, is null within a uniform spherical shell, acts on a particle outside of a uniform spherical shell (or a body made up of a series of uniform spherical shells) as if the mass of that shell (or body made up of shells) were concentrated at its geometric center, has a value proportional to the distance from the center within a uniform sphere, and so on.

Such mathematical specifications of causes are different from physical explanations of the origin and mode of action of causes. This leads us to a recognition of the hierarchy of causes which it is important to keep in mind in understanding the specific qualities of the Newtonian revolution in science. For instance, Kepler found that planets move in ellipses with the sun at one focus, and that a line drawn from the sun to a planet sweeps out equal areas in equal times. Both of these laws encompass actual observations within a mathematical framework. The area law enabled Kepler to account for (or to explain) the nonuniformity of the orbital motion of planets, the speed being least at aphelion and greatest at perihelion. This is on the level of a mathematical explanation of the nonuniform motion of planets. Kepler, however, had gone well beyond such a mathematical explanation, since he had assigned a physical cause for this variation by supposing a celestial magnetic force; but he was never successful in linking this particular force mathematically to the elliptical orbits and the area law, or in finding an independent phenomenological or empirical demonstration that the sun does exert this kind of magnetic force on the planets (see Koyré, 1973, pt. 2, sect. 2, ch. 6; Aiton, 1969; Wilson, 1968).

Newton proceeded in a different manner. He did not begin with a discussion of what kind of force might act on planets. Rather he asked what are the mathematical properties of a force—whatever might be its causes or its mode of action, or whatever kind of force it might be—that can produce the law of areas. He showed that, for a body with an initial component of inertial motion, a necessary and sufficient condition for the area law is that the said force be centripetal, directed continually toward the point about which the areas are reckoned. Thus a mathematically descriptive law of motion was shown by mathematics to be equivalent to a set of causal conditions of forces and motions. Parenthetically it may be observed that the situation of a necessary and sufficient condition is rather unusual; most frequently it is the case that a force or other "cause" is but a sufficient condition for a given effect, and in fact only one of a number of such possible sufficient conditions. In the *Principia* the conditions of central forces and equal areas in equal times lead to considerations of elliptical orbits, which were shown by Newton to be a consequence of the central force varying inversely as the square of the distance (see Ch. 5).

Newton's mathematical argument does not, of course, show that in the orbital motion of planets or of planetary satellites these bodies are acted on by physical forces; Newton only shows that within the conceptual framework of forces and the law of inertia, the forces acting on planets and satellites must be directed toward a center and must as well vary inversely as the square of the distance. But in the hierarchy of causal explanation, Newton's result does finally direct us to seek out the possible physical properties and mode of action of such a centrally directed inverse-square force.^{11} What is important in the Newtonian mode of analysis is that there is no need to specify at this first stage of analysis what kind of force this is, nor how it acts. Yet Newton's aim was ultimately to go on by a different mode of analysis from the mathematical to the physical properties of causes (or forces) and so he was primarily concerned with 'verae causae', causes-as he said-that are 'both true and sufficient to explain the phenomena'.^{12}

This hierarchy of mathematical and physical causes may be seen also in Newton's analysis of Boyle's law, that in a confined gas (or "elastic fluid", as it was then called) the pressure is inversely proportional to the volume. We have seen that Boyle himself suggested two alternative physical explanations of the spring of the air in relation to his law, but declined to declare himself in favor of either of them. In the *Principia* (as we shall see in 3.3) Newton showed that, on the supposition that there is a special kind of force of mutual repulsion between the particles composing such an "elastic fluid", Boyle's law is both a necessary and sufficient condition that this force vary inversely as the distance. Again there is a hierarchy of mathematical and physical analyses of cause. In this second Newtonian example, it is more obvious that the physical conditions assumed as the cause of the law are themselves open to question. Newton himself concluded his discussion of this topic (*Principia,* bk. two, prop. 23) by observing that it is 'a physical question' as to 'whether elastic fluids [i.e., compressible gases] do really consist of particles so repelling one another'. He himself had been concerned only with the mathematical demonstration, so he said, in order that natural philosophers (or physical scientists) might discuss the question whether gases may be composed of particles endowed with such forces. With regard to the hierarchy of mathematical and physical causes, there is of course no real formal difference between the Newtonian analysis of Kepler's laws and of Boyle's law. In the case of Kepler's laws, however, Newton could take the law of inertia for granted, as an accepted truth of the new science, so that there would have to be some cause for the planets to depart from a rectilinear path and to trace out an elliptical orbit. If this cause is a force, then it must be directed toward a point (the sun, in the case of the planets), since otherwise there can be no area law. But in the case of compressible gases or elastic fluids, the situation is somewhat different. In the first place, in Newton's mind there was no doubt whatsoever that such 'elastic fluids do really consist of particles', since he was a firm believer in the corpuscular philosophy; but it is to be observed that there were many scientists at that time who, like the followers of Descartes, believed in neither atoms nor the void. But even if the particulate structure of gases could be taken for granted, there was the additional property attributed to such particles by Newton, that they be endowed with forces which enable them to repel one another. Many of those who believed in the "mechanical philosophy" and accepted the doctrine of particularity of matter would not necessarily go along with Newton's attribution of forces to such particles, whether atoms, molecules, or other forms of corpuscles. Furthermore, as Newton makes plain in the scholium which follows his proposal of an explanatory physical model for Boyle's law, 'All these things are to be understood of particles whose centrifugal forces terminate in those particles that are next to them, and are diffused not much further.' Accordingly, there is a great anid wide gulf between the supposition of a set of mathematical conditions from which Newton derives Boyle's law and the assertion that this is a physical description of the reality of nature. As will be explained in Ch. 3, it is precisely Newton's ability to separate problems into their mathematical and physical aspects that enabled Newton to achieve such spectacular results in the *Principia.* And it is the possibility of working out the mathematical consequences of assumptions that are related to possible physical conditions, without having to discuss the physical reality of these conditions at the earliest stages, that marks the Newtonian style.

The goal of creating an exact physical science based on mathematics was hardly new in the seventeenth century. O. Neugebauer has reminded us that Ptolemy, writing in the second century A.D., had declared this very aim in the original title of his great treatise on astronomy, which we know as the *Almagest,* but which he called 'Mathematical Composition' (or 'Compilation') (Neugebauer, 1946, p. 20; cf. Neugebauer, 1948, pp. 1014-1016). But there was a fundamental difference between the old and the new mathematical physical science, which may be illustrated by an aspect of planetary theory and the theory of the moon.

It is well known that in the *Almagest* Ptolemy was concerned to produce or develop geometric models that would serve for the computation of the latitudes and longitudes of the seven "planetary bodies" (the five planets plus sun and moon) and hence would yield such special information as times of eclipses, stationary points, conjunctions and oppositions. These were quite obviously mathematical models, and were not intended to partake of physical reality. Thus there was no assumption that the true motion of these planetary bodies in the heavens necessarily is along epicycles moving around on deferents and controlled by equal angular motion about an equant. In particular, Ptolemy was perfectly aware that his order of the planets (in terms of increasing distance from the earth: moon, Mercury, Venus, sun, Mars, Jupiter, Saturn) was somewhat arbitrary for the five "planets", since their distances cannot be determined by parallaxes. In fact, Ptolemy admits that some astronomers would place Mercury and Venus beyond the sun, while others would have Mercury on one side of the sun and Venus on the other.^{13} Again, in the theory of the moon, Ptolemy introduced a "crank" mechanism, which would increase the 'apparent diameter of the epicycle' so as to make the model agree with positional observations. As a result Ptolemy was able to make an accurate representation of the moon's motion in longitude, but only at the expense of introducing a fictitious variation in the distance of the moon from the earth, according to which 'the apparent diameter of the moon itself should reach almost twice its mean value, which is very definitely not the case' (Neugebauer, 1957, p. 195). This departure from reality was one of the telling points of criticism raised by Copernicus in his *De revolutionibus* (1543). Descartes also proposed hypothetical models that, according to his own system, had to be fictitious.

Newton believed that he had proved that gravity, the cause of terrestrial weight and the force producing the downward acceleration of freely falling bodies, extends as far as the moon and is the cause of the moon's motion. He gave a series of arguments that it is this same force that keeps the planets in their orbits around the sun and the satellites in their orbits about their respective planets. He showed, furthermore, how this force of gravity can account for the tides in the seas and the irregularities (as well as the regularities) in the moon's motion. He set forth the goal of explaining the moon's motion in a new way—not by celestial geometry and models which (like Ptolemy's) obviously cannot possibly correspond to reality, but rather by 'true causes' ('verae causae') whose properties could be developed mathematically. Thus Newtonian theory would reduce the features of the moon's motion to two sources: the interactions of the earth and the moon, and the perturbing effects of the sun. It is to be remarked that this procedure does not depend on the origin, nature, or physical cause of the gravitating force but only on certain mathematically elucidated properties, as that this force varies inversely as the square of the distance, that it is null within a spherical shell (or within a homogeneous sphere or a sphere made up of homogeneous shells), that the action of a sphere on an external particle is the same as if all the mass of the sphere were concentrated at its geometric center, that within a solid sphere the force on a particle is as the distance from the center, and so on. Such investigations did not depend on whether a planet is pushed or pulled toward the center, whether gravitation arises from an aether of varying density or a shower of aether particles or is even a simple action-at-a-distance. For Newton these latter questions were far from irrelevant to a complete understanding of the system of the world, and we know that he devoted considerable energy to them. Furthermore, the mathematical analysis had revealed some of the basic properties of the force and thus made precise the analysis into its cause. But in Newton's hierarchy of causes, the elucidation of the properties of universal gravity was distinct from—that is, on a different level from—the search for the cause of gravity. He thus put forth the radical point of view in the concluding General Scholium to the *Principia:* It is enough ('satis est') that gravity exists and that it acts according to the laws he had mathematically demonstrated, and that this force of gravity suffices 'to explain all the motions of the heavenly bodies and of our sea' (see n. 12 *supra).* How revolutionary this proposal was can be seen in the number of scientists and philosophers who refused to accept it and who rejected the *Principia* together with its conclusions because they did not approve of the concept of "attraction".

**
1.5 Causal mathematical science in the Scientific Revolution
**

In the last section, an outline was given of a hierarchy in the mathematical science of nature. On the lowest or most primitive level, this phrase may mean no more than mere quantification or calculation. Numerical data may provide arguments to test or to buttress essentially nonmathematical theories such as Harvey's. On a simple level, primarily in the realms of physics and astronomy, mathematics came to signify not only the measurements of positions and apparent (observed) angular speeds, and the rather straightforward application of plane and spherical trigonometry to the solutions of problems of the celestial sphere, but also the increasing quantification of qualities ranging from temperature to speeds. The ideal was to express general laws of nature as mathematical relations between observable physical quantities, notably in relation to the science of motion: kinematics and then dynamics. Such laws expressed number-relations or geometrical properties, and they were formalized in ratios or proportions, algebraic equations (or their equivalents in words), together with geometric properties and trigonometric relations, and eventually the infinitesimal calculus and other forms of higher mathematics, notably infinite series.

Since such mathematical laws use physically observable quantities (volume, weight, position, angle, distance, time, impact, and so on), they can to a large degree be tested by further observations and direct experiment, which may limit the range in which they hold: examples are Boyle's, Snel's, and Hooke's law, and the forms of Kepler's law of refraction.' Or, the test may be the verification or nonverification of a prediction (as the occurrence or nonoccurrence of a lunar or solar eclipse or a particular planetary configuration), or the accurate retrodiction of past observations. Obviously, some kind of numerical data must provide the basis for applying or testing such general or specific mathematical laws or relations. In all of this, there is and there need be no concern for physical causes. Galilean science is a foremost example of the successful application of mathematics to physical events on this level. Cause enters in the argument only to the extent of an awareness that air resistance may cause a slowing down of an otherwise uniform rectilinear motion (or component of motion), and that weight may cause an acceleration downward. Thus, for Galileo, motion could continue uniformly in a straight line only if there were no air resistance and if there were an extended horizontal plane to support the mobile, on which it could move without friction.^{2}

We have seen, however, that in the seventeenth century there were found to be significant quantitative laws that cannot be directly tested, such as the law of uniform motion for falling bodies that speeds acquired are as the times elapsed (*v*_{1}:*v*_{2} = *t*_{1}:*t*_{2}). Galileo, as we saw, could do no more than confirm another law of falling bodies, that the distances are in the squared ratio of the times (*s*_{1}:*s*_{2} = (*t*_{1}:*t*_{2})^{2}); and then, since the distance law is a consequence of the speed law, he supposed that the truth (verified by experiment) of the distance law guaranteed the truth of the speed law. In our modern language, the testability of *s* ∝ *t*^{2} is the way to confirm *v* ∝ *t*. This is a classic and simple example of what has generally come to be called the hypothetico-deductive method. Galileo tested the distance-time relation for the accelerated motion upon an inclined plane, for various angles of inclination, and showed that *s* does maintain a constant proportion to *t*^{2}. Since this relation was an inference (or deduction) from an assumption (or hypothesis) that *v* is proportional to *t*, the hypothetico-deductive method assumes that experimental confirmation of the deduced result *s*_{1}:*s*_{2} = *t*_{1}:*t*_{2}^{2} guarantees the validity of the hypothesis *v*_{1}:*v*_{2} = *t*_{1}:*t*_{2} from which the relation of *s* to *t*^{2} had been deducel (see §1.4). As Ernst Mach (1960, p. 161) put it, in his celebrated *Science of Mechanics,* 'The inference from Galileo's assumption was thus confirmed by experiment, and with it the assumption itself The limitations to this mode of confirmation are twofold. One is philosophical: are there any ways to be sure that only *v* ∝ *t* implies *s* ∝ *t*^{2}? That is, granted that *v* ∝ *t* is a sufficient condition for *s* ∝ *t*^{2}, is it also a necessary condition?^{3} The second is historical as well as philosophical. That is, a scientist may make an error in logic or mathematics: This is illustrated by the fact that at one stage in his career Galileo believed that the verifiable relations *s*_{1}:*s*_{2} = (*t*_{1}:*t*_{2})^{2} follows from the speeds being proportional to distances (*v*_{1}:*v*_{2} = *s*_{1}:*s*_{2}) rather than the relation of speeds to times (*v*_{1}:*v*_{2} = *t*_{1}:*t*_{2}) (see Galileo, 1974, pp. 159sq; 1890-1909, vol. 8, p. 203).

The Galilean science of motion embodies only one part of the revolution in the exact sciences in the seventeenth century. For, in addition to the production of exact or mathematical laws, systems, and general constructs that may or may not be like models that conform to the direct experience of nature (experiment and observation), there arose the ideal of finding the true physical causes of such laws, systems, constructs, and models, in a hierarchy of causes that began with the mathematical elucidation of the properties of forces causing motions and only then proceeded to the analysis of the nature and cause of such forces.^{4} The degree to which this goal was first achieved in Newton's *Principia* set the seal on an accomplished Scientific Revolution and was in and of itself revolutionary. Lest my readers should suppose that this is an anachronistic judgment of the twentieth century superimposed upon the events of the past, let me anticipate here one aspect of the next chapter, by indicating that this was an unequivocal judgment in the Age of Newton. Clairaut, Newton's immediate intellectual successor in celestial mechanics, declared unambiguously in 1747, 'The renowned treatise of Mathematical principles of Natural Philosophy [of Isaac Newton] inaugurated a great revolution in physics', a sentiment echoed by Lagrange and others (Clairaut, 1749; see §2.2).

The program for this revolution in physical science was first clearly set forth in astronomy, in the declared goal to put aside all noncausal and nonphysical computing schemes and to discover how the sun, moon, and planets actually move in relation to the physical ("true") causes of their motions. This aspect of the revolution found its first major spokesman in Kepler, whose *Astronomia nova* (1609), or *Commentary on the Motion of Mars,* was also described by him as a 'physica coelestis', a celestial physics (see Caspar, 1959, pp. 129sqq; Koyré, 1973, pp. 166sqq, 185sqq). What made this work 'new' was that it was not merely an *Astronomia nova,* but an *Astronomia nova* &i-rtoXoyiytos, a 'new astronomy based on causes'; and this was the sense in which Kepler declared it to be a 'celestial physics'.' That is, Kepler was not content with the limited goal of previous astronomers (including Ptolemy, Copernicus, and Tycho Brahe) of choosing a fitting center of motion and then determining planetary motions by judicious combinations of circular motions that would 'save the phenomena' (cf. Duhem, 1969). He wanted to derive planetary motions from their causes, from the forces that are the causes of the motions. He thus rejected one of the basic aspects of Copernican astronomy: that planetary orbits be computed with respect to an empty point in space corresponding to the center of the earth's orbit, rather than the sun itself. Kepler reasoned that forces originate in bodies, not in points in space; hence the motion of the planets must be reckoned in relation to the center of planetary force, the central body, the sun. As a result, Kepler attempted a dynamical rather than a kinematical astronomy, based on laws of force and motion rather than on applied geometry and arithmetic (see Koyré, 1973; Cohen, 1975*a*; Beer & Beer, 1975, sect. 10). Certain of Kepler's fellow astronomers disapproved of his thus introducing into astronomy a set of physical causes and hypotheses; it were better (said his former teacher, Michael Maestlin) to stick to the traditional geometry and arithmetic (letter to Kepler, 21 Sept. 1616; Kepler, 1937-, vol. 17, p. 187). Of course, it was easier to effect this radical change in Kepler's day than it would have been earlier, since Tycho Brahe had effectively demonstrated that comets move through the solar system. As Tycho himself put it: Had there ever existed crystalline spheres to which the planets were attached, they were now shattered and existed no longer. Hence, for anyone who went along with Tycho's conclusions, there was need for a wholly new scheme for explaining how the planets can possibly move in their observed curved paths.^{6}

And so we are not surprised to find that Descartes also sought for causal explanations of the celestial motions, and so did certain other astronomers of the early seventeenth century, such as Bullialdus and Borelli.^{7} But others were content to confine their attention more nearly to the phenomenological level of prediction and observation, without exhibiting any concern as to physical causes, or the possible reality (or lack of reality) of geometric computational schemes. From this point of view, one of the most astonishing aspects of Galileo's *Dialogue concerning the Two Chief World Systems* is the absence of any celestial physics. Galileo, in fact, seems never to have concerned himself with any speculations on the possible forces that might be acting in the operation of a Copernican system.^{8} In this sense, Galileo was not at all a pioneer in celestial mechanics, as Kepler and Descartes were, however much his personal contributions to the science of motion influenced the course of development of theoretical dynamics at large. But he was concerned with the truth and reality of the Copernican system, and he even advanced an explanation of the tides that seemed to him to require that the earth rotate about its axis while revolving around the sun.

The enormous advance in the exact physical sciences in the seventeenth century may be gauged by the gap between both Galileo's kinematics and Kepler's faulty and unsuccessful dynamics^{9} on the one hand and Newton's goal of a mathematical dynamics congruent with the phenomenological kinematical laws and disclosing their physical cause on the other. Kepler, who in so many of his precepts resembles Newton, nevertheless represents a wholly different level of scientific belief and procedure. Kepler starts out from the causes, whereas Newton concludes in causes. Kepler accepts a kind of celestial attraction, based on an analogy with terrestrial magnetism, and seeks its consequences; Newton arrives at his concept of universal gravitation only after the logic of the study of forces and motions leads him in that direction (see Ch. 5). Newton's philosophy directs him from effects to causes, and from particulars to generalities. But Kepler believed it best to proceed in the reverse direction. 'I have no hesitation', he wrote, 'in asserting that everything that Copernicus has demonstrated *a posteriori* and on the basis of observations interpreted geometrically, may be demonstrated *a priori* without any subtlety or logic'.^{10}

Like Galileo's laws of falling bodies, Kepler's laws were shown by Newton to be true only in rather limited circumstances which Newton then actually specified. Newton sought to determine new forms of these laws that would be more universally true. As we shall see in Ch. 3, the revolutionary power of Newton's method came from his ability to combine new modes of mathematical analysis with the study of physical causes, controlled constantly by the rigors of experiment and observation. But an essential ingredient was Newton's clear recognition of the hierarchy of causes and his ability to separate the mathematical laws from the physical properties of forces as causes. In this process he did not produce mere mathematical constructs or abstractions that were devoid of any content of reality other than "saving the phenomena", but he did create what he conceived to be purely mathematical counterparts of simplified and idealized physical situations that could later be brought into relation with the conditions of reality as revealed by experiment and observation. It was this aspect of Newtonian science, in my opinion, that produced so outstanding a result that his *Principia* was conceived to have been or to have inaugurated the epoch of a revolution in science, or at least to have brought to a level of revolutionary fruition the goals of creating a mathematical science of nature that had been expressed, however imperfectly, by Galileo and by Kepler.

**
Notes
**

*General note:* The extracts from Newton's *Principia* and *System of the World* are either from new translations (in progress) by I. B. Cohen and Anne Whitman or are generally revisions of existing translations.

§1.1

**
1. The Newtonian revolution in science
**

^{1} In recent years, much of the discussion concerning scientific revolutions has centered on Kuhn (1962). For comments on Kuhn's views, see Lakatos & Musgrave (1970). For a modified statement of Kuhn's views, see his paper, 'Second thoughts on paradigms', in Suppe (1974). The propriety of using the term "revolution" to describe scientific change is denied in Toulmin (1972), vol. 1, pp. 96-130, esp. pp. 117sq.

^{2} This expression is used by historians in a generally uncritical manner that does not necessarily imply adherence to any particular concept of revolution or even to a specific and clearly formulated doctrine of historical change. On the history of this concept and name, see Ch. 2 and Cohen (1977*e*).

^{3} Pierre Duhem was largely responsible for the view that many of the discoveries traditionally attributed to Galileo had been anticipated by late medieval thinkers. Duhem's thesis concerning the medieval origin of modern science has been put forth in a new way in Crombie (1953).

^{4} Still valuable are such older works as Ornstein (1928), the only comprehensive work ever produced on this subject, and Brown (1934); to be supplemented by such recent works as Hahn (1971), Middleton (1971), and Purver (1967).

^{5} The official name of the Royal Society is: The Royal Society of London for Improving Natural Knowledge.

^{6} On the history of scientific journals, see Thornton & Tully (1971), and Knight (1975), esp. ch. 4.

^{7} For Bacon's views on utility, see his *Novum organum,* bk. 1, aph. 73, aph. 124; bk. 2, aph. 3. Descartes's statements concerning the ways in which science can make us 'the masters and possessors, as it were, of nature' (chiefly 'the conservation of health' and 'the invention of … artifacts which would allow us the effortless enjoyment of the fruits of the earth …') are to be found in pt. 6 of his *Discours de la méthode* (chiefly the end of the second and the beginning of the third paragraphs, and toward the conclusion).

^{8} Two now-classic statements concerning social influences on seventeenth-century science are given in Hessen (1931) and Merton (1938). A stunning example of how to investigate 'the place of science within the conceptual framework of economic, social, political and religious ideas' has been given for the period in Webster (1975), and cf. the long and thoughtful review of Webster's work by Quentin Skinner, *Times Literary Supplement* (2 July 1976), no. 3877, pp. 810-812.

^{9} On Newton and ship design, see Cohen (1974*b*); on the longitude at sea, see Newton (1975), introduction, pt. 5; concerning the telescope, see Newton (1958), sect. 2, §§3-5 (§17 contains a description of another instrument, a reflecting octant for use in practical navigation, which was found among Newton's papers and which he never saw fit to make public).

^{10} Descartes, in his *Discours de la méthode* (1637), says explicitly that he does not consider himself to be a man of more than average mental capacity; hence if he has done anything extraordinary, the reason must be that he had a sound method (see Descartes, 1956, p. 2).

^{11} Galileo's own account occurs in the fifth paragraph of the text of his *Sidereus nuncius* (1610); Galileo (1890-1909), vol. 3, pt. 1, pp. 60sq; trans. by Drake in Galileo (1957), p. 29.

^{12} Newton (1672), p. 3075; reprinted in facs. in Newton (1958), p. 47. See Newton (1959-1977), vol. 1, p. 92.

^{13} The exception was Domingo de Soto (*d.* 1560), who in a commentary on Aristotle's *Physics* (1545) 'was the first to apply the expression "uniformly difform" to the motion of falling bodies, thereby indicating that they accelerate uniformly when they fall and thus adumbrating Galileo's law of falling bodies'. Quoted from William A. Wallace's (1975) account of Soto in the *Dictionary of Scientific Biography.* See Beltrán de Heredia (1961); Wallace (1968); Clagett (1959), pp. 257, 555sq, 658.

^{14}*Principia,* bk. one, prop. 4, corol. 6, corol. 7. For other examples of Newton's considerations of mathematical relations that do not occur in nature, see §3.3.

^{15} In the scholium following prop. 78 (bk. one) of the *Principia,* Newton refers to these two as 'major cases of attractions' and finds it 'worthy of note' that under both conditions, the attractive force of a spherical body follows the same law as that of the particles which compose it. On this topic see §3.1, n. 5.

^{16} In reaction, Alexandre Koyré came to the opposite conclusion: that, far from relying on experiment (and being the founder of the modern experimental method), Galileo was not primarily an experimenter. Furthermore, Koyré even asserted that many of Galileo's most celebrated experiments could not have been performed, at least not in the manner described. See Koyré (1943), (1950*a*), and (1960*c*); these are collected in Koyré (1968). On Galileo's thought-experiments, see Shea (1972), pp. 63-65, 156, 157sq. It is generally recognized today that Koyré's point of view was extreme and needs some modification. Some of the "unperformable" experiments cited by Koyré have since then been performed and yield the very results described by Galileo; see Settle (1961), (1967), and MacLachlan (1973). Drake has recently found that experiment very likely played a significant role in Galileo's discoveries of the principles of motion.

§1.2

^{1} Leibniz and Newton both had a share in this revolution (see n. 2 to §2.2). But it must be kept in mind that Newton made a large number of discoveries or inventions in mathematics, among them the general binomial expansion of *(a + b)n,* the fundamental theorem that finding the area under a curve and finding the tangent to a curve are inverse operations, the methods of both the differential and integral calculus, the classification of cubic curves, various properties of infinite series, both the Taylor and Maclaurin expansions, modes of calculation and methods of numerical analysis (including the methods of successive iteration, interpolation, etc.), plus other aspects of geometry, analysis, and algebra. On these topics see Whiteside's introduction to Newton (1964-1967), and Whiteside's introductions and running commentary to his edition of Newton's *Mathematical Papers* (Newton, 1967-).

^{2} Newton's public positive contributions to chemistry are conveniently summed up in Partington (1961), ch. 13.

^{3} The contents of these queries are conveniently summarized in Duane H. D. Roller's analytical table of contents in the Dover edition of the *Opticks* (Newton, 1952, pp. lxxix-cxvi) and in Cohen (1956), pp. 164-171, 174-177. On the development of the Queries, see Koyré (1960*c*).

^{4} On this tract see Newton (1958), pp. 241-248, 256-258; also (1959-1977), vol. 3, pp. 205-214.

^{5} All extracts from Newton's *Principia* are given in the text of a new translation, now in progress, by I. B. Cohen and Anne M. Whitman, or are revisions of Andrew Motte's translation.

^{6} From Newton's unpublished *Conclusio,* trans. by A. R. & M. B. Hall (1962), p. 333.

^{7} See Roller's analytical table of contents (Newton, 1952, pp. lxxix-cxvi).

^{8} Young (1855), vol. 1, pp. 161, 183sq; see Peacock (1855), pp. 150-153. But it should not be thought that every number given in the published *Opticks* represents an exact measurement or the result of a computation based on such direct measurement.

^{9} For example, Roberts & Thomas (1934) is subtitled 'A study of one of the earliest examples of scientific method' and is part of a series with the general title, Classics of Scientific Method.

^{10} The strictly methodological portions of the *Opticks* are to be found in the final paragraph of qu. 28 and in the concluding pages of the lengthy qu. 31 with which the second English edition of 1717/1718 concludes; both had appeared in earlier versions in the Latin edition of 1706.

^{11} Quoted from Dover ed. (Newton, 1952), pp. 369sq. This query was first published in the Latin edition (1706) as qu. 20 and then appeared in revised form in English in the second English edition (1717/1718).

^{12} Quoted from Dover ed. (Newton, 1952), pp. 404sq. This query also appeared first in the Latin edition (as qu. 23) and then in revised form in the second English edition.

^{13} The word "scenario" is used because Newton wrote up his initial experiments with prisms in an apparently autobiographical manner, but his MSS hint that he was trying to impose a Baconian and experimental-inductivist scenario upon what must have been the sequence of his prior beliefs, experiences, and conclusions. On this subject see Lohne (1965), (1968); Sabra (1967), pp. 245-250.

^{14}" See §3.6, n. 5. These "phenomena" were called "hypotheses" in the first edition of the *Principia;* see Koyré (1955*b*), Cohen (1966).

^{15} This is made evident in a table I have prepared for a commentary on the *Principia* (in progress), in which I have tabulated the occurrence of every explicit reference to a definition or law, as well as to a preceding proposition or lemma, and—in the case of bk. three—the rules, phenomena, and hypotheses.

^{16} The question of "analysis" and "synthesis" may cause real confusion in relation to Newton's scientific work. This pair of words (of Greek origin) and their counterparts (of Latin origin) "resolution" and "composition" are used by Newton in a general scientific sense and specifically, in qu. 31 of the *Opticks,* to describe how 'by this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them' and 'in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general'. Then 'the Synthesis consists in assuming the Causes discovered, and established as Principles, and by them explaining the Phaenomena proceeding from them, and proving the Explanations'. Newton also refers to the 'Two Methods of doing things' of 'Mathematicians … which they call Composition & Resolution'.

Long ago Dugald Stewart showed that "analysis" and "synthesis" have different meanings in mathematics and in physics and that Newton therefore speaks with some imprecision in apparently relating the modes of investigation in physics or natural philosophy and in mathematics. Stewart even shows how in some cases "analysis" and "synthesis" may have opposing senses in the two realms. See Stewart's *Elements of the Philosophy of the Human Mind,* ch. 4 ('Logic of induction'), sect. 3 ('Of the import of the words analysis and synthesis in the language of modern philosophy'), subsect. 2 ('Critical remarks on the vague use, among modern writers, of the terms *analysis* and *synthesis');* Stewart (1877), vol. 3, pp. 272sqq.

For a recent example of an attempt to make the method of analysis and synthesis, as expounded in qu. 31 of the *Opticks,* apply to the *Principia,* see Guerlac (1973). For Newton's published statement on analysis and synthesis in mathematics, and in the *Principia,* see n. 21 *infra.*

^{17} Turbayne (1962), pp. 46, 49. On the geometric form of presentation in essentially nonmathematical books, see §3.11.

^{8} The *Principia* was first written as two "books" *(De motu corporum).* Newton then expanded the end of the first "book" into a second "book" (on motion in resisting fluids, pendulum motion, wave motion, etc.), calling *these* two "books" *De motu corporum.* The subject matter of the original second "book" was wholly recast and became the third "book" of the *Principia* (called Liber tertius, *De mundi systemate).* After Newton's death, the text of the original bk. two was published in Latin and in an English version called respectively *De mundi systemate liber* (London, 1728) and *A Treatise of the System of the World* (London, 1728; London, 1731). It is in the English version of this latter work that the famous phrase, 'in a mathematical way', appears. See my introduction to Newton (1975), p. xix.

^{9} Cuvier (1812), 'Discours preliminaire', p. 3. 'Sans doute les astronomes ont marche plus vite que les naturalistes, et l'epoque ou se trouve aujourd'hui la theorie de la terre, ressemble un peu a celle ou quelques philosophes croyoient le ciel de pierres de taille, et la lune grande comme le Peloponese:mais apres les Anaxagoras, il est venu des Copemic et des Kepler, qui ont fraye la route a Newton; et pourquoi l'histoire naturelle n'auroit-elle pas aussi un jour son Newton?'

According to John T. Edsall (personal communication), Otto Warburg, in discussing the problem of biological oxidations around 1930, said: 'Heute, wie vor fuinfzig Jahren, gilt das van't Hoffsche Wort: Der Newton der Chemie is noch nicht gekommen.' No doubt Warburg was referring to the general introduction ('An die Leser') to vol. 1 of the *Zeitschrift fur Physikalische Chemie* (Leipzig, 1887), p. 2, where the state of chemistry is compared with the condition of astronomy in 'Kopernikus' und Kepler's Zeit', and the need is expressed for a 'Newton der Chemie'. This preface was signed jointly by Van't Hoff and Ostwald.

^{20} A major exception is the general scholium, at the end of sect. 6, bk. two (which was at the end of sect. 7 in the first edition), and the scholium at the end of sect. 7 (first published in the second edition). The latter scholium describes Newton's investigations of the resistances of fluids by experiments on bodies falling in air and in water. The general scholium is devoted to Newton's experiments on the resistances of fluids, in which he studied the oscillations of pendulums under various conditions and compared the motion of pendulums in air, water, and mercury.

^{21} In his later life, Newton attempted to superimpose on the history of the *Principia a* chronology in which he would have developed and used the new fluxional calculus in an algorithmic form so as to have discovered the main propositions by analysis, and then have recast them in the form of Greek geometry according to the method of synthesis. Thus he wrote: 'By the help of the new Analysis Mr. *Newton* found out most of the Propositions in his *Principia Philosophiae:* but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the Systeme of the Heavens might be founded upon good Geometry. And this makes it now difficult for unskilful Men to see the Analysis by which those Propositions were found out'; quoted from Newton (1715), p. 206; cf. Cohen (1971), p. 295. There is no documentary evidence whatever to support this scenario, while abundant evidence favors the view that Newton's mode of discovery follows more or less the form of presentation in the published *Principia.*

§1.3

^{1} On Hales see Guerlac (1972); Cohen (1976*b*); F. Darwin (1917), pp. 115-139.

^{2} A contemporaneous English translation of Van Helmont's account of his experiment is given in Partington (1961), p. 223.

^{3}*De motu cordis,* ch. 9; quoted from Harvey (1928). Cf. Kilgour (1954) and especially Pagel (1967), pp. 73sqq.

^{4} Hence the quantitative method used by Harvey is at least as revolutionary as his conclusions about the circulation, and possibly even more so. From a seventeenth-century point of view, such subjects as theoretical statics, kinematics, and dynamics were exact mathematical sciences which became physical sciences only when applied to physics.

^{5} Galileo (1974), p. 147; (1890-1909), vol. 8, p. 190. Strictly speaking, Galileo did not ever express his physical laws as the algebraic proportions *s* ∝ *t ^{2}* or v ∝

*t.*As a matter of fact it is misleading even to write out his results in the form

*s*

_{1}:

*s*

_{2}=

*t*

_{1}

^{2}:

*t*

_{2}

^{2}, much less (

*s*

_{1}/

*s*

_{2}) = (

*t*

_{1}/

*t*

_{2}). On this point see n. 8

*infra.*In what foilows, I shall (as a kind of shorthand) refer to relations that Galileo found as

*s*∝

*t*

^{2}or

*v*∝

*t*, but without any intended implication that these are Galileo's formulations of such laws.

^{6} 'On naturally accelerated motion', third day, prop. 2; Galileo (1974), p. 166; (1890-1909), vol. 8, p. 209.

^{7} In corol. I to prop. 2, Galileo shows that although the total distance traversed is proportional to the square of the time, the distances traversed in each successive equal interval of time are as the odd numbers starting from unity—a result that follows from number theory, since the sequence 1, 4, 9, 16, 25, … leads to the sequence 1(= I 0), 3(= 4 - 1), 5(= 9 - 4), 7(= 16 - 9), 9(= 25 - 16), …

^{8} Galileo did not restrict himself to such number relations. Thus (third day, prop. 2 on accelerated motion: 1974, p. 166; 1890-1909, vol. 8, p. 209): 'If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times'. Galileo's proportion is thus space_{1}:space_{2} = (time_{1}:time_{2})^{2}; he does not use the functional relation *s* á *t*^{2}. On this point see the comments by Drake in the introduction to Galileo (1974), pp. xxi-xxiv. But in discussing quantities of the same kind (e.g., line segments), Galileo would use verbal equivalents of equations, such as *"HB* est excessus *NE* super *BL" (HB* = *NE BL).*

^{9} Quoted from Il *saggiatore* ('The assayer'), sect. 6, trans. in Crombie (1969), vol. 2, p. 151; Galileo (1890-1909), vol. 6, p. 232. This statement is omitted in Drake's version, Galileo (1957). I do not wish to enter here into discussions of Galileo's possible Platonism, for which see Koyré (1943) and a rebuttal by Clavelin (1974). Geymonat (1965), pp. 198sq, warns against reading this particular remark of Galileo's out of context.

^{10} Newton's *Principia* is apt to be described, on the basis of superficial examination, as a treatise in the style of Greek geometry. Although the extemal form displays a geometrical style reminiscent of Euclid, a closer examination shows that Newton's method is not at all like that of the classic Greek geometers; rather, proposition by proposition and lemma by lemma, he usually proceeds by establishing geometrical conditions and their corresponding ratios and then at once introducing some carefully defined limiting process. In sect. I of bk. one, Newton sets forth general principles of limits (which he calls the method of first and last ratios), so that he may apply some degree of rigor to problems using nascent or evanescent quantities (or ratios of such quantities) in the rest of the treatise. Furthermore, even in the matter of ratios and proportions, Newton is a "modern"; he does not follow the Greek style, in the sense that Galileo does and that Kepler tends to do. That is, he writes "mixed" proportions, implying a direct functional relationship, for instance stating that a force may be directly or inversely proportional to some condition of a distance. Traditionally one would have had to say that one force is to another as a condition of some distance is to that same condition of another distance. Finally, one of the distinctive features of the *Principia,* as noted by Halley in his review, was the extensive and innovative use of the method of infinite series, which shows the degree to which the *Principia* is definitely not a treatise in Greek geometry. On this topic see Cohen (1974*c*), pp. 65sqq, and Whiteside (1970*b*).

^{11} Kepler's system of nested spheres is delineated in his *Mysterium cosmographicum* (1596; rev. ed. 1621). An annotated English version of this work has been completed by Eric J. Aiton and Alistair M. Duncan (Kepler, in press).

In a letter of 1595, Kepler said: 'The world of motion must be considered as made up of rectilinear [regular solids]. Of these, however, there are five. Hence if they are to be regarded as the boundaries or partitions … they can separate no more than six objects. Therefore six movable bodies revolve around the sun. Here is the reason for the number of planets'; quoted from Kepler (1965), p. 63. This example shows how considerations of shape and geometry were not necessarily free of numerical aspects.

^{12} In the *Narratio prima* (or *First Account),* trans. in Rosen (1971), p. 147, Rheticus said, 'What is more agreeable to God's handiwork than that this first and most perfect work should be summed up in this first and most perfect number?' For a history of this problem, see Cohen (1977*d*).

^{13} Quoted from Kepler's *Mysterium cosmographicum* (1596) in Kepler (1937-), vol. 1, p. 9; trans. in Rufus (1931), p. 10. Cf. Koyré (1973), p. 128.

^{14}" Sizi found other grounds for his assertion: the seven primary metals in alchemy, the time when the embryo starts to form in a mother's womb (seven hours after coitus), the date at which a human fetus is sufficiently alive to survive if born prematurely (seven months after conception). See Drake (1958) and Ronchi's introduction to Clelia Pighetti's translation of Sizi (1964).

^{15} Kepler (1965), p. 14. Since the earth has one satellite and Jupiter has four, a geometric sequence would yield two for Mars, eight for Saturn, and none for Mercury and Venus. The number 6, which Kepler suggests as an alternative for the eight satellites attributed to Saturn, is more difficult to account for. We may well understand, accordingly, why some scholars have made a "silent correction" of this 6, so as to have it be 7 or 5. This number 6 does fit the arithmetic progression 2,4,6, but in this case the earth would have no satellite, which would actually negate the basis of assigning two to Mars. Furthermore, Kepler also suggested that Venus and Mercury might have one satellite each. These two numbers would break the sequence, but there could be no other choice if each planet is to have no more satellites than the immediately superior planet.

^{16} For details see Cohen (1977*b*), (1977*d*).

^{17} For other examples of numerology in nineteenth- and twentieth-century science, see Cohen (1977*d*). An outstanding example is the so-called Bode's law (or the Titius-Bode law), which gives reasonably good values for the planetary distances (up to Uranus), including a place for the asteroids. It too fails for the first term. See, further, Nieto (1972).

§1.4

^{1} Quoted from *Mysterium cosmographicum* (1596) in Kepler (1937-), vol. 1, p. 9; trans. in Rufus (1931), p. 9. Cf. Koyré (1973), p. 138. In fact, Kepler went on to say that he 'was induced to try and discover them [i.e., these three things] because of the wonderful resemblance between motionless objects, namely, the sun, the fixed stars and intermediate space, and God the Father, God the Son, and God the Holy Ghost; this analogy I shall develop further in my cosmography'.

^{2}*Mysterium cosmographicum,* ch. 20. To see how this law works, note that the periods of Mercury and Venus are respectively 88^{d} and 224 ^{d}; hence half the increase in period is ½(224 ^{d} - 88^{d}) = 68 ^{d}. Kepler's rule is that 88:(88 + 68) = dist. of Mercury:dist of Venus. The results, as given by Dreyer (1906), p. 379, are Jupiter:Saturn 0.574 (0.572), Mars:Jupiter 0.274 (0.290), earth:Mars 0.694 (0.658), Venus:earth 0.762 (0.719), Mercury:Venus 0.563 (0.500); the number in parentheses in each case is the Copernican value. In an equation, Kepler's rule would read (*T*_{n} + *T*_{n-1})*/2T*_{n-1} = *A*_{n}*/A*_{n-1}; cf. Koyré (1973), pp. 153sq."

^{3} See Kepler's *Harmonice mundi,* in Kepler (1937-), vol. 6, p. 302. Kepler is astonishingly silent as to how he came upon this law. Koyré (1973), p. 455, n. 27, discusses some conjectures on this topic made by J. B. Delambre and R. Small (he gives his own opinion on p. 339). See, further, Gingerich (1977).

^{4} This difference between the third law and the first two is seen in Newton's treatment of them. He admits that the third law, 'found by Kepler, is accepted by everyone' (hypoth. 6, ed. 1, phen. 4, eds. 2 and 3, *Principia,* bk. three); but he does not give Kepler credit in the *Principia* for either the area law or the law of elliptical orbits, and at least once he claimed that Kepler had only guessed planetary orbits 'to be elliptical'.

^{5} Galileo's Platonic "discovery" did not embody a "causal" explanation in the sense of assigning a physical cause to the supposed accelerated motions of the planets toward the sun: for example, by supposing forces that might be operative in the celestial system to produce such accelerations.

^{6} Cf. Koyré (1960*b*), reprinted in Koyré (1965), where (p. 218, n. 3) Koyré discusses how A. E. Taylor believed (erroneously, as it turned out) that he had found the source of this supposed cosmological doctrine of Plato.

^{7} Cf. Cohen (1967*c*). Newton also pointed out other faults in Galileo's assumptions; see Whiteside's commentary in Newton (1967-), vol. 6, pp. 56sq, n. 73.

^{8} Galileo was aware that in projectile motion there is an acceleration in the same direction as gravity or weight; whereas there is no acceleration or deceleration (save for the slight retardation caused by air resistance) at right angles to that downward direction. This is not, however, a real anticipation (however limited) of the second law, since Galileo does not specify clearly that the "impeto" is an external force acting on a body in order to produce an acceleration. And the same is true of Galileo's analysis of motion along an inclined plane, where both the "impeto" of gravity and the acceleration are diminished in the ratio of the sine of the angle of elevation. Drake has discussed Galileo's concept of cause in his introduction to Galileo (1974), pp. xxvii-xxviii; see, further, Drake (1977).

^{9} For a convenient summary of the medieval physics of motion see Grant (1971), ch. 4. For texts and translations see Clagett (1959) and Grant (1974), sects. 40-51.

^{10} For Galileo's description of this series of experiments, see Galileo (1974), pp. 169sq; (1890-1909), vol. 8, pp. 212sq.

^{11} Whereas Kepler begins with the nature of the force, Newton concludes in the inquiry into the nature of a force with certain properties that have come to light during the antecedent investigations: as that it diminishes with the square of the distance, extends to great distances, and is proportional to the mass of bodies, etc.

^{12} Concluding general scholium to the *Principia:* see, further, §3.2.

^{13}*Almagest,* bk. 9, sect. 1.1. In his *Planetary Hypotheses,* Ptolemy developed a physical system or a physical model of astronomy in addition to the mathematical computing models of astronomy described in the *Almagest.* Cf. Hartner (1964), supplemented by Goldstein (1967).

§1.5

^{1} A doctoral dissertation on Kepler's optics was completed in 1970 by Stephen Straker (Indiana University).

^{2} Galileo was aware that if a moving body were to continue in motion along a horizontal path (tangent to the earth), it would in effect be getting farther and farther away from the earth's center or surface: rising up, as it were, *sponte sua.*

^{3} Galileo himself fell into this trap in his arguments for the Copemican system. He developed a theory in which the tides are produced by a combination of the motions of the earth. Hence, he believed (and argued), God must have created the universe with the earth rotating on its axis and revolving in an orbit, just as Copernicus said. Pope Urban VIII argued against the "conclusiveness" of this proof of the Copernican system on the grounds that it would limit God's omnipotence. Galileo had only shown that his version of the Copernican system would imply tidal phenomena similar to those we observe; he had not demonstrated the converse. His Copernican system was a sufficient condition to explain the tides, but it was not a necessary condition. On Galileo's theory of the tides, see Aiton (1954) and Burstyn (1962); also Aiton (1963) and Burstyn (1963).

^{4} Of course, as we shall see below in Ch. 4, a system, construct, or model could successively gain additional features that would bring it so nearly into harmony with experience that it would seem to be a description of reality.

^{5} Koyré (1973), p. 166, says that 'the very title of Kepler's work proclaims, rather than foretells, a revolution'.

^{6} Both Kepler and Newton held that the demise of the concept of crystalline spheres required a theory of planetary motions based on forces.

^{7} On Borelli, see Koyré (1952*a*). There is no adequate study of the celestial system of Descartes or of Bullialdus.

^{8} In Galileo's presentation of Plato's cosmological scheme (see §1.4, nn. 5 and 6, as well as Galileo, 1953, pp. 29sq; 1890-1909, vol. 4, pp. 53sq; also 1974, pp. 232-234, and 1890-1909, vol. 8, pp. 283sq), he seems to have assumed that when a planet was started off in its orbit with the proper speed, it would then move in its orbit without needing the action of any force.

^{9} Unsuccessful and faulty as a system in general, Kepler's dynamics did, however, serve to establish the first two Keplerian laws of planetary motion. See Koyré (1973), pp. 185-244; Krafft (1973).

^{10}*Mysterium cosmographicum* (1596), quoted in Duhem (1969), p. 101; Kepler (1937-), vol. 1, p. 16.

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